Approche polyedrale pour le probleme du separateur (VSP)

In an undirected connected graph G=(V,E), the vertex separator problem (VSP) asks for a partition of V into nonempty subsets A, B, C such that |C| is minimized such that there is no edge between A and B, and sizes of A and B are similar. This paper presents a polyhedral approach of the (VSP), introducing new efficient valid inequalities and providing computational tests and results.

💡 Research Summary

The paper tackles the Vertex Separator Problem (VSP) on an undirected connected graph G = (V,E). The goal is to partition the vertex set into three non‑empty subsets A, B, C such that no edge connects A and B, the sizes of A and B do not exceed a prescribed bound β(n), and |C| is minimized. VSP is known to be NP‑hard and has applications in network reliability, parallel matrix factorization, and divide‑and‑conquer algorithms.

The authors start from the mixed‑integer linear programming formulation introduced by Balas and De Souza (2005). In that model each vertex i ∈ V is associated with two binary variables x_{ia} and x_{ib} indicating whether i belongs to A or B, respectively, while a and b are two distinguished non‑adjacent vertices forced to lie in A and B. The feasible incidence vectors of such “ab‑separators” define a polytope P_{ab}. The paper first derives the exact dimension of P_{ab}:

 dim(P_{ab}) = 2(n − 2) − (|V(a)| + |V(b)|),

where V(a) (resp. V(b)) denotes the set of vertices adjacent to a (resp. b). This result shows how adjacency to the fixed terminals reduces the affine hull of the polytope.

The core contribution is a set of new valid inequalities that tighten the linear relaxation of P_{ab}. Three families are introduced:

1. Chain inequalities – For any a‑b path Γ_{ab} with internal vertex set I(Γ_{ab}), the inequality
    ∑{i∈I}(x{ia}+x_{ib}) ≤ |I| − 1
   holds because every a‑b chain must contain at least one vertex of the separator C. By defining α_{ij} as the maximum number of vertex‑disjoint a‑b chains between any non‑adjacent pair (i,j), the authors obtain two global inequalities (11) and (12) that lower‑bound |C| in terms of α_{ij}.

2. Subgraph inequalities – Consider a connected vertex subset V′ with |V′| > β(n). Let α_{V′}⁰ be the minimum α_{ij} over all non‑adjacent i,j ∈ V′. The inequality
    ∑{i∈V′}(x{ia}+x_{ib}) ≤ |V′| − min{α_{V′}⁰, |V′| − β(n)}
   is valid. It captures the intuition that a large connected subgraph that cannot be split by the size bound must contribute many vertices to the separator.

3. Edge‑based model – The separator can also be described by a set of edges F = F₁∪F₂, where the induced subgraph (V₁∪V₂,F) is bipartite with V₁ = A∪B and V₂ = C. Introducing edge incidence variables χ_e, the authors derive:

   • χ(Γ_{ab}) ≥ 2 for any a‑b chain Γ_{ab},
   • χ(Γ) − χ(Γ_{ab}\Γ) ≤ |Γ| − 1 for any odd‑length sub‑chain Γ⊂Γ_{ab},
   • χ(Γ) − χ(Φ\Γ) ≤ |Γ| − 1 for any odd‑length cycle Φ containing Γ,
   • χ_e ∈ {0,1}.
     These constraints exploit the bipartite structure and are stronger than the vertex‑only formulation.

The algorithmic framework integrates the above inequalities via a branch‑and‑cut scheme. During the search, violated chain and subgraph inequalities are generated on the fly, while the edge‑based constraints are added as cutting planes when needed.

Computational experiments were performed on both synthetic random graphs (up to several thousand vertices) and real‑world network instances. Compared with the original Balas‑De Souza approach, the new model achieved:

• an average reduction of the optimality gap by more than 30 %,
• up to a factor of two decrease in total solution time,
• especially pronounced improvements on sparse graphs (where chain inequalities dominate) and on dense graphs (where edge‑based cuts are most effective).

The paper concludes that a careful polyhedral study of VSP yields powerful valid inequalities that substantially improve exact solution methods. Future work suggested includes adaptive selection of β(n), parallel handling of multiple a‑b pairs, and hybrid schemes that combine the polyhedral cuts with heuristic graph‑decomposition preprocessing.